Heuristic Parameter Choice in Tikhonov Method from Minimizers of the Quasi-Optimality Function

Raus, Toomas; Hämarik, Uno

doi:10.1007/978-3-319-70824-9_12

Cited by 9 publications

(11 citation statements)

References 28 publications

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“…Thus, all the known results for the quasi-optimality principle extend to the weakly-bounded noise case. For advanced numerical implementations of this method, see [17].…”

Section: The Quasi-optimality Rulementioning

confidence: 99%

Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

Kindermann

Raik

2019

Numerical Functional Analysis and Optimization

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We study the choice of the regularisation parameter for linear ill-posed problems in the presence of noise that is possibly unbounded but only finite in a weaker norm, and when the noise-level is unknown. For this task, we analyse several heuristic parameter choice rules, such as the quasi-optimality, heuristic discrepancy, and Hanke-Raus rules and adapt the latter two to the weakly bounded noise case. We prove convergence and convergence rates under certain noise conditions. Moreover, we analyse and provide conditions for the convergence of the parameter choice by the generalised cross-validation and predictive mean-square error rules.

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“…Thus, all the known results for the quasi-optimality principle extend to the weakly-bounded noise case. For advanced numerical implementations of this method, see [17].…”

Section: The Quasi-optimality Rulementioning

confidence: 99%

Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

Kindermann

Raik

2019

Numerical Functional Analysis and Optimization

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“…The study of heuristic methods of selecting the regularization parameter is still reasonable (see [14,15,12,5]) because of the importance and frequency of situations where no delta estimation is available.…”

Section: Introduction Consider An Ill-posed Linear Equation (11)mentioning

confidence: 99%

“…The heuristic discrepancy rule. The regularization parameter α = α HD is chosen as the global minimizer of the particular simple functional [7,9,15]). A modification of this very simple method will be further investigated for the discrete problem.…”

Section: Introduction Consider An Ill-posed Linear Equation (11)mentioning

confidence: 99%

A new heuristic parameter choice rule in Tikhonov regularization applied for Ritz approximation of an ill-posed problem

Regińska¹

2021

Appl. Math. (Warsaw)

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The paper concerns a certain parameter selection criterion for a combination of Ritz approximation of an ill-posed problem and Tikhonov regularization in the case when the level of data noise is unknown. A new minimization-based noise-level-free choice rule is considered. The new functional depends on the discretization level, the Tikhonov regularization parameter, and noisy data. The functional is minimized over an interval depending on a number expressing how well the discrete operator approximates the initial one. In order to obtain the convergence of the discrete regularized solutions to the exact solution when the data noise tends to 0, a certain specific qualitative assumption on the noise is introduced. The convergence is proved under source conditions on the exact solution, the noise condition and certain additional assumptions on the operator approximation.

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“…with relatively small constant c and we show that at least one parameter from set L min has this property. For the choice of proper parameter from the set L min some algorithms were proposed in [34], in the current work we propose other algorithms. We propose to construct Q-curve which is the analogue of the L-curve [18], but on x-axis we use modified discrepancy instead of discrepancy and on the y-axis the function ψ Q (α) instead of u α .…”

Section: Introductionmentioning

confidence: 99%

Q-Curve and Area Rules for Choosing Heuristic Parameter in Tikhonov Regularization

Raus

Hämarik

2020

Mathematics

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We consider choice of the regularization parameter in Tikhonov method if the noise level of the data is unknown. One of the best rules for the heuristic parameter choice is the quasi-optimality criterion where the parameter is chosen as the global minimizer of the quasi-optimality function. In some problems this rule fails. We prove that one of the local minimizers of the quasi-optimality function is always a good regularization parameter. For the choice of the proper local minimizer we propose to construct the Q-curve which is the analogue of the L-curve, but on the x-axis we use modified discrepancy instead of discrepancy and on the y-axis the quasi-optimality function instead of the norm of the approximate solution. In the area rule we choose for the regularization parameter such local minimizer of the quasi-optimality function for which the area of the polygon, connecting on Q-curve this minimum point with certain maximum points, is maximal. We also provide a posteriori error estimates of the approximate solution, which allows to check the reliability of the parameter chosen heuristically. Numerical experiments on an extensive set of test problems confirm that the proposed rules give much better results than previous heuristic rules. Results of proposed rules are comparable with results of the discrepancy principle and the monotone error rule, if the last two rules use the exact noise level.

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Heuristic Parameter Choice in Tikhonov Method from Minimizers of the Quasi-Optimality Function

Cited by 9 publications

References 28 publications

Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

A new heuristic parameter choice rule in Tikhonov regularization applied for Ritz approximation of an ill-posed problem

Q-Curve and Area Rules for Choosing Heuristic Parameter in Tikhonov Regularization

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